Scheduling jobs with a general learning effect model

The paper deals with machine scheduling problems with a general learning effect. By the general learning effect, we mean that the actual processing time of a job is not only a non-increasing function of the total weighted normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence, where the weight is a position-dependent weight. We show that even with the introduction of a general learning effect to job processing times, some single machine scheduling problems are still polynomially solvable under the proposed model. We also show that some special cases of the flow shop scheduling problems can be solved in polynomial time.     

Single Machine Batch Scheduling Problem with Resource Dependent Setup and Processing Time in the Presence of Fuzzy Due Date

We consider a batch scheduling problem on a single machine which processes jobs with resource dependent setup and processing time in the presence of fuzzy due-dates given as follows:1. There are n independent non-preemptive and simultaneously available jobs processed on a single machine in batches. Each job j has a processing time and a due-date.2. All jobs in a batch are completed together upon the completion of the last job in the batch. The batch processing time is equal to the sum of the processing times of its jobs. A common machine setup time is required before the processing of each batch.3. Both the job processing times and the setup time can be compressed through allocation of a continuously divisible resource. Each job uses the same amount of the resource. Each setup also uses the same amount of the resource.4. The due-date of each job is flexible. That is, a membership function describing non-decreasing satisfaction degree about completion time of each job is defined.5. Under above setting, we find an optimal batch sequence and resource values such that the total weighted resource consumption is minimized subject to meeting the job due-dates, and minimal satisfaction degree about each due-date of each job is maximized. But usually we cannot optimize two objectives at a time. So we seek non-dominated pairs i.e. the batch sequence and resource value, after defining dominance between solutions.A polynomial algorithm is constructed based on linear programming formulations of the corresponding problems.     

Dynamic routing to heterogeneous collections of unreliable servers

We argue the importance of problems concerning the dynamic routing of tasks for service in environments where the servers have diverse characteristics and are subject to breakdown. We propose a general model in which both service times and repair times at each machine are i.i.d.with some general distribution. Routing decisions take account of queue lengths, machine states (up or down), the elapsed processing times of jobs in service and the times to date of any machine repairs in progress. We develop an approach to machine calibration which yields a machine index which is a function of all of the preceding information. The heuristic which routes all tasks to the machine of current smallest index performs outstandingly well. The approach of the paper is flexible and is capable of yielding strongly performing routing policies for a range of variants of the basic model. These include cases where job processing is lost at each breakdown and where the machine state may be only partially observed. AMS Subject Classifications 49L20, 90B22     

Single machine batch scheduling with resource dependent setup and processing times

Jobs are processed by a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time common for all batches. Both the job processing times and the setup time can be compressed through allocation of a continuously divisible resource. Each job uses the same amount of the resource. Each setup also uses the same amount of the resource, which may be different from that for the jobs. Polynomial time algorithms are presented to find an optimal batch sequence and resource values such that either the total weighted resource consumption is minimized, subject to meeting job deadlines, or the maximum job lateness is minimized, subject to an upper bound on the total weighted resource consumption. The algorithms are based on linear programming formulations of the corresponding problems.     

Single machine group scheduling with resource dependent setup and processing times

A single machine scheduling problem is studied. There is a partition of the set of n jobs into g groups on the basis of group technology. Jobs of the same group are processed contiguously. A sequence independent setup time precedes the processing of each group. Two external renewable resources can be used to linearly compress setup and job processing times. The setup times are jointly compressible by one resource, the job processing times are jointly compressible by another resource and the level of the resource is the same for all setups and all jobs. Polynomial time algorithms are presented to find an optimal job sequence and resource values such that the total weighted resource consumption is minimum, subject to meeting job deadlines. The algorithms are based on solving linear programming problems with two variables by geometric techniques.     

Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects

In this paper we consider the single machine past-sequence-dependent (p-s-d) setup times scheduling problems with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects, we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem, maximum tardiness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.     

Range of lower bounds

Each of n jobs is to be processed without interruption on a single machine. Each job becomes available for processing at time zero. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times added with maximum weighted tardiness. In this paper we give a general case of the theorem that given in [6]. This theorem shows a relation between the number of efficient solutions, lower bound LB and optimal solution. It restricts the range of the lower bound, which is the main factor to find the optimal solution. Also, the theorem opens algebraic operations and concepts to find new lower bounds.     

An approximation algorithm for parallel machine scheduling with a common server

In this paper we study the scheduling of a given set of jobs on several identical parallel machines tended by a common server. Each job must be processed on one of the machines. Prior to processing, the server has to set up the relevant machine. The objective is to schedule the jobs so as to minimize the total weighted job completion times. We provide an approximation algorithm to tackle this intractable problem and analyze the worst-case performance of the algorithm for the general, as well as a special, case of the problem.     

Learning effect and deteriorating jobs in the single machine scheduling problems

This paper studies the single machine scheduling problems with learning effect and deteriorating jobs simultaneously. In this model, the processing times of jobs are defined as functions of their starting times and positions in a sequence. It is shown that even with the introduction of learning effect and deteriorating jobs to job processing times, the makespan, the total completion time and the sum of the k$k$th power of completion times minimization problems remain polynomially solvable, respectively. But for the following objective functions: the total weighted completion time and the maximum lateness, this paper proves that the shortest weighted processing time first (WSPT) rule and the earliest due-date first (EDD) rule can construct the optimal sequence under some special cases, respectively.     

Batch scheduling with controllable setup and processing times to minimize total completion time

We study a problem of scheduling n jobs on a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time dependent on the position of this batch in the batch sequence. Setup times and job processing times are continuously controllable, that is, they are real-valued variables within their lower and upper bounds. A deviation of a setup time or job processing time from its upper bound is called a compression. The problem is to find a job sequence, its partition into batches, and the values for setup times and job processing times such that (a) total job completion time is minimized, subject to an upper bound on total weighted setup time and job processing time compression, or (b) a linear combination of total job completion time, total setup time compression, and total job processing time compression is minimized. Properties of optimal solutions are established. If the lower and upper bounds on job processing times can be similarly ordered or the job sequence is fixed, then O(n³ log n) and O(n⁵) time algorithms are developed to solve cases (a) and (b), respectively. If all job processing times are fixed or all setup times are fixed, then more efficient algorithms can be devised to solve the problems.     

Single machine scheduling with a general exponential learning effect

In this paper we consider the scheduling problem with a general exponential learning effect and past-sequence-dependent (p-s-d) setup times. By the general exponential learning effect, we mean that the processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the δ ? 0th power of completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Single machine scheduling with general time-dependent deterioration, position-dependent learning and past-sequence-dependent setup times

The paper deals with single machine scheduling problems with setup time considerations where the actual processing time of a job is not only a non-decreasing function of the total normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We consider the following objective functions: the makespan, the total completion time, the sum of the δth (δ ≥ 0) power of job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the δ th (δ ≥ 0) power of job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Group Scheduling with Controllable Setup and Processing Times: Minimizing Total Weighted Completion Time

The following single machine scheduling problem is studied. A partition of a set of n jobs into g groups on the basis of group technology is given. The machine processes jobs of the same group contiguously, with a sequence independent setup time preceding the processing of each group. The setup times and the job processing times are controllable through the allocation of a continuously divisible or discrete resource to them. Each job uses the same amount of the resource. Each setup also uses the same amount of resource, which may be different from that for the jobs. Polynomial-time algorithms are constructed for variants of the problem of finding an optimal job sequence and resource values so as to minimize the total weighted job completion time, subject to given restrictions on resource consumption. The algorithms are based on a polynomial enumeration of the candidates for an optimal job sequence and solving the problem with a fixed job sequence by linear programming. This research was supported in part by The Hong Kong Polytechnic University under grant number G-T246 and the Research Grants Council of Hong Kong under grant number PolyU 5191/01E. In addition, the research of M.Y. Kovalyov was supported by INTAS under grant number 00-217.     

一个非常规目标函数的单机随机调度的最优策略

本文考虑了n个工件在同一台机器上加工的调度问题 ,其中工件的加工时间和交货期都是具有任意分布的随机变量 .我们考虑了一个非常规目标函数 ,其中工件的权数与平均加工时间成比例 .在工件的交货期与加工时间满足相容条件下 ,得到了个简单的最优排序策略 .     

工件存在不同交货宽容期的最大加权超前延误问题

本文考虑下述排序问题：有n个工件需在同一台机器上加工,对各工件有一宽容交货期,若一工件在其宽容期前完工则受加权超前惩罚,若在其宽容期后完工则受加权延误惩罚,要求适当安排一加工方式使最大惩罚最小,文中相应某指定工件需准时完工的上述问题证得了Np-hard性,给出了最优算法,并作了一些讨论。     

ON-LINE PROBLEMS OF MINIMIZING MAKESPAN ON A SINGLE BATCH PROCESSING MACHINE WITH NONIDENTICAL JOB SIZES

The on-line problem of scheduling on a batch processing machine with nonidentical job sizes to minimize makespan is considered. The batch processing machine can process a number of jobs simultaneously as long as the total size of these jobs being processed does not exceed the machine capacity. The processing time of a batch is given by the longest processing time of any job in the batch. Each job becomes available at its arrival time, which is unknown in advance, and its processing time becomes known upon its arrival. The paper deals with two variants： the case only with two distinct arrival times and the general case. For the first case, an on-line algorithm with competitive ratio 119/44 is given. For the latter one, a simple algorithm with competitive ratio 3 is given. For both variants the better ratios can be obtained if the problem satisfies proportional assumption.     

Asymptotical optimality of WSEPT for stochastic online scheduling on uniform machines

We study the stochastic online scheduling on m uniform machines with the objective to minimize the expected value of total weighted completion times of a set of jobs that arrive over time. For each job, the processing time is a random variable, and the distribution of processing time is unknown in advance. The actual processing time could be known only when the job is completed. For the problem, we propose a policy which is proved to be asymptotically optimal when the processing times and weights are uniformly bounded, i.e. the relative error of the solution achieved by our policy approaches zero as the number of jobs increases to infinity.     

一个宽容交货超前延误单机排序问题

此文考虑下述排序问题（P）：有n个工件需在同一台机器上加工，对各工件有一共同的宽容交货期。若一工件在此宽容期前完工则为一超前工件，若在此宽容期后完工则为一延误工件，要求适当安排一加工方式和宽容交货期的位置使加权超前延误工件数量小。文中证得（P）是NP-hard的，并给出一伪多项式时间的分枝状精确算法，这也就可以认为它是一般意义下的NP-hard问题而不是强NP-hard问题。     

具有不可用区间且工件可拒绝下的单机重新排序问题的近似方案

本文考虑了机器具有不可用区间且工件可拒绝下的单机重新排序问题,在该问题中,给定一个工件集需在一台机器上加工,每个工件有自己的加工时间和权重,且对该工件集目标函数为极小化总加权完工时间的排序计划已给定,根据该排序计划中每个工件的完工时间已确定每个工件的承诺交付时间。然而,在工件正式开始加工前,原计划用于加工的某段时间区间因临时用于检修机器而导致机器在该时间区间不再可用,需要对工件重新排序。为了确保在新的重新排序中,工件的延误成本不致太大,决策者可以选择拒绝部分工件,但需支付相应的拒绝费用。任务是确定接受工件集和拒绝工件集,并将接受的工件在考虑机器具有不可用区间的条件下重新排序使得接受工件集的总加权完工时间,总拒绝费用及赋权最大延误之和最小。该问题是NP-困难的,对此给出了伪多项式时间动态规划精确算法,利用稀疏技术设计了完全多项式时间近似方案。